Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-24T14:09:58.987Z Has data issue: false hasContentIssue false
  • English
  • Français

Topological and symbolic dynamics for hyperbolic systems with holes

Published online by Cambridge University Press:  17 November 2010

STEFAN BUNDFUSS ,
TYLL KRÜGER  and
SERGE TROUBETZKOY
STEFAN BUNDFUSS
Affiliation:
Department of Mathematics, Technische Universität Darmstadt, Germany (email: [email protected])
TYLL KRÜGER
Affiliation:
FB Mathematik, Technische Universität Berlin and Fakultät für Physik, Universität Bielefeld, Germany (email: [email protected])
SERGE TROUBETZKOY
Affiliation:
Centre de Physique Théorique, Fédération de Recherche des Unités Mathématiques de Marseille, Institut de Mathématiques de Luminy, and Université de la Méditerranée, Marseille, France (email: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider an axiom A diffeomorphism or a Markov map of an interval and the invariant set Ω* of orbits which never falls into a fixed hole. We study various aspects of the symbolic representation of Ω* and of its non-wandering set Ωnw. Our results are on the cardinality of the set of topologically transitive components of Ωnw and their structure. We also prove that Ω* is generically a subshift of finite type in several senses.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 5 , October 2011 , pp. 1305 - 1323
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]van den Bedem, H. and Chernov, N.. Expanding maps of an interval with holes. Ergod. Th. & Dynam. Sys. 22 (2002), 637–654.CrossRefGoogle Scholar
[2]Blanchard, F.. β-shifts and symbolic dynamics. Theoret. Comput. Sci. 65 (1989), 131–141.CrossRefGoogle Scholar
[3]Blanchard, F. and Hansel, G.. Systèmes codés. Theoret. Comput. Sci. 44 (1986), 17–49.CrossRefGoogle Scholar
[4]Blanchard, F. and Hansel, G.. Sofic constant-to-one extensions of subshifts of finite type. Proc. Amer. Math. Soc. 112 (1991), 259–265.CrossRefGoogle Scholar
[5]Bruin, H., Demers, M. and Melbourne, I.. Existence and convergence properties of physical measures for certain dynamical systems with holes. Ergod. Th. & Dynam. Sys 30(3) (2010), 687–728.CrossRefGoogle Scholar
[6]Bunimovich, L. and Yurchenko, A.. Where to place a hole to achieve a maximal escape rate, 2008, arXiv:0811.4438.Google Scholar
[7]de Carvalho, A.. Pruning fronts and the formation of horseshoes. Ergod. Th. & Dynam. Sys. 19 (1999), 851–894.CrossRefGoogle Scholar
[8]Chernov, N. and Markarian, R.. Ergodic properties of Anosov maps with rectangular holes. Bol. Soc. Bras. Mat. 28 (1997), 271–314.CrossRefGoogle Scholar
[9]Chernov, N. and Markarian, R.. Anosov maps with rectangular holes. Nonergodic cases. Bol. Soc. Bras. Mat. 28 (1997), 315–342.CrossRefGoogle Scholar
[10]Chernov, N., Markarian, R. and Troubetzkoy, S.. Conditionally invariant measures for Anosov maps with small holes. Ergod. Th. & Dynam. Sys. 18 (1998), 1049–1073.CrossRefGoogle Scholar
[11]Chernov, N., Markarian, R. and Troubetzkoy, S.. Invariant measures for Anosov maps with small holes. Ergod. Th. & Dynam. Sys. 20 (2000), 1007–1044.CrossRefGoogle Scholar
[12]Collet, P., Martinez, S. and Schmitt, B.. The Yorke–Pianigiani measure and the asymptotic law on the limit Cantor set of expanding systems. Nonlinearity 7 (1994), 1437–1443.CrossRefGoogle Scholar
[13]Cvitanović, P.. Dynamical averaging in terms of periodic orbits. Phys. D 83 (1995), 109–123.CrossRefGoogle Scholar
[14]Demers, M., Wright, P. and Young, L.-S.. Escape rates and physically relevant measures for billiards with small holes. Comm. Math. Phys. 294(2) (2010), 353–388.CrossRefGoogle Scholar
[15]Demers, M. and Young, L.-S.. Escape rates and conditionally invariant measures. Nonlinearity 19 (2006), 377–397.CrossRefGoogle Scholar
[16]Demers, M.. Markov extensions and conditionally invariant measures for certain logistic maps with small holes. Ergod. Th. & Dynam. Sys. 25 (2005), 1139–1171.CrossRefGoogle Scholar
[17]Demers, M.. Markov extensions for dynamical systems with holes: an application to expanding maps of the interval. Israel J. Math. 146 (2005), 189–221.CrossRefGoogle Scholar
[18]Falconer, K.. The Geometry of Fractal Sets (Cambridge Tracts in Mathematics, 85). Cambridge University Press, Cambridge, 1985.CrossRefGoogle Scholar
[19]Gaspard, P. and Dorfman, J.. Chaotic scattering theory, thermodynamic formalism, and transport coefficients. Phys. Rev. E 52 (1995), 3525–3552.CrossRefGoogle ScholarPubMed
[20]Keller, G. and Liverani, C.. Rare events, escape rates and quasistationarity: some exact formulae. J. Stat. Phys. 135 (2009), 519–534.CrossRefGoogle Scholar
[21]Lind, D. and Marcus, B.. Symbolic Dynamics and Coding. Cambridge University Press, Cambridge, 1995.CrossRefGoogle Scholar
[22]Liverani, C. and Maume-Deschamps, V.. Lasota–Yorke maps with holes: conditionally invariant probability measures and invariant probability measures on the survivor set. Ann. Inst. H. Poincaré Probab. Statist. 39 (2003), 385–412.CrossRefGoogle Scholar
[23]Lopes, A. and Markarian, R.. Open billiards: Cantor sets, invariant and conditionally invariant prbabilities. SIAM J. Appl. Math. 56 (1996), 651–680.CrossRefGoogle Scholar
[24]Pianigiani, G. and Yorke, J.. Expanding maps on sets which are almost invariant: decay and chaos. Trans. Amer. Math. Soc. 252 (1979), 351–366.Google Scholar
[25]Ruelle, D.. Entropy production in nonequilibrium statistical mechanics. Comm. Math. Phys. 189 (1997), 365–371.CrossRefGoogle Scholar